IRREVERENT QUANTUM MECHANICS Giancarlo Borgonovi May 2004

MOTIVATION

What is irreverent quantum mechanics? A discipline for OFs to keep involved with QM: Develop allegories/metaphors about QM Design/build models/representations of QM effects Investigate QM trivia Explore connection between science and art Write fiction around QM subjects/characters Develop humor about QM subjects/characters Quantum mechanical cooking? Give presentations to other OFs.

GENERAL PRINCIPLES

Classical and quantum mechanics comparison QuantumClassical System State vector Represented by real numbers Possible states Definite state Deterministic transition from one state to another System State vector Represented by complex numbers Possible states Superposition of states Probabilistic transition from one state to another

The formal elements of quantum mechanics Abstract state vector Abstract state vector in dual space Probability amplitude for going from state A to state B Matrix element of operator Operator

The great law of quantum mechanics From The Feynman Lectures on Physics, Vol. 3

That is a statement, not a question I have not understood how you passed from A to B Are there any questions? The unforgiving logic of P. A. M. Dirac

Observables in Quantum Mechanics Represented by real operators Describe possible states (eigenvectors) which are associated with possible outcomes of measurements (eigenvalues) Before the measurement: calculate probabilities of different outcomes After the measurement: only one outcome Example Expectation values for different cases

? Hilbert space and human life

Human life according to Classical Mechanics Hamilton’s Equations

Human life according to Quantum Mechanics Schroedinger Equation

Schroedinger Heisenberg DiracFeynman The different forms of quantum mechanics Wave FunctionMatrix Mechanics Symbolic MethodPath Integral A B

Max Planck, studying the black body radiation, discovers the “brick ”. Planck’s constant h = 6.55 x erg sec can be considered as the building block of quantum mechanics. h

h 2π = A new, downsized model of the ‘brick’ is introduced

The ‘brick’ is split in half (Uhlenbeck and Goudsmit introduce the spin).

Particles position and momentum and Heisenberg uncertainty principle

BOSONS and FERMIONS

A wrong representation of the hands of God building matter A more realistic representation of the hands of God building matter Identical particles are not distinguishable

Quantum Mechanics divides the Universe into two Categories Every particle in the universe is either a boson or a fermion, that is to say everything in the universe is made up of bosons and fermions. What distinguishes a boson from a fermion? What are the effects of this categorization?

What distinguishes a boson from a fermion 1) Bosons have spin integer, fermions have spin semi-integer 2) The possible states for a system of bosons (at least two) are symmetric 3) The possible states for a system of fermions (at least two) are antisymmetric 4) Two bosons interfere with the same phase 5) Two fermions interfere with the opposite phase.

+ + Boson + - Fermion + - Pauli or ExclusionPrinciple Shapes represent quantum states, colors represent particles ( Symmetric under exchange) ( Antisymmetric under exchange) ( Null for fermions under exchange)

Effects due to boson like features Bosons are very gregarious and tend to congregate together. If bosons exist in a state, there is a tendency for another boson to enter that state. The laser is an example of this tendency of the bosons to come together Superfluidity of Helium-4 (not Helium-3 which emulates a fermion) at low temperature is a macroscopic example of the result of the tendency of bosons to get into the same state of motion.

Effects due to fermion like features Fermions tend to avoid each other. If a fermion exists in a state, another fermion will not want to enter that state. Pauli’s Exclusion Principle What if electrons were bosons

Electrons as fermions (real) Electrons as bosons (imagined) Matter under different assumptions From The Feynman Lectures on Physics, Vol. 3

Classical and Quantum Statistics

Bosons Fermions Fermi sphere The different nature of bosons and fermions My army of bosons will move and attack as one man Unknown Barbarian King Everyone in my army of fermions will occupy his place and defend the empire Unknown Roman Emperor

New States of Matter

THE PERIODIC TABLE (Ability and Weirdness)

Quantum Mechanics and Weirdness - Thoughts about the periodic table

FORMATION OF THE PERIODIC TABLE

Low Angular Momentum High Angular Momentum Spherical symmetry, angular momentum, and weirdness

Sociological implications of the periodic table Consider the order of the states as some kind of social order, or rank, or job position. In a rigid, hierarchical society, positions would be occupied according to certain parameters (e.g. diplomas, family connections, religious or ethnical factors, etc.). In a more intelligent society, people of higher ability pass in front of others and acquire a higher social status. This process has some similarity to the buildup of the periodic table. Thus nature rewards ability. The external shells, which are responsible for the chemical behavior of the elements, consist of s and p electrons only. The “weirder” d and f electrons are left behind, and are used to fill incomplete shells, so in a sense they hide behind less weird electrons at a higher level. Thus, nature tends to hide weirdness.,

SECOND QUANTIZATION and QUANTUM FIELDS

Second Quantization Occupation number representation This operator creates or destroys particles Fixed number of particles

One- particle space (Hilbert space} N- particle space Many particle space (Fock space) Symmetric or antisymmetric states Collection of n-particle states Principle of symmetrization QUANTUM MECHANICAL SPACES

VIRTUAL PARTICLES Virtual particles are like words, they can result in attraction or repulsion Virtual particles have a very short lifetime An exchange of momentum can be interpreted as the action of a force over a time interval Photons Electromagnetic field PhononsCooper pairs, superconductivity MesonsNucleons GluonsQuarks Hideki Yukawa

Quantum Fields  A classical field is easy to visualize and understand  A quantum field is an operator which is a function of position  To understand a quantum field one needs to understand the local creation and annihilation operators  Everything (energy, number of particles, total momentum, etc.) can be expressed in terms of the creation and annihilation operators  A quantum field is expressed in terms of creation and annihilation operators  A quantum field is a nice way to express the duality particle wave that pervades QM  What are the eigenvalues and eigenvectors of a quantum field?

Quantum Cooking - Potatoes a la Brillouin Leon Brillouin, 1927

THANK YOU AND MAY YOU HAVE A HAPPY TRANSITION TO A STATE OF HIGHER ANGULAR MOMENTUM